Buried electronic markers are used to indicate the location of a buried structure or utility. A buried marker is made from a circular coil that is arranged in a resonant circuit and designed to resonate at a specific frequency. An oscillatory electric current may be induced in this circuit by an externally applied pulse or pulses of magnetic flux linking the coil. The oscillatory current in the coil gives rise to an oscillatory magnetic field around the coil. The presence of this oscillatory magnetic field may be detected, allowing the position of the marker to be determined. The axis of the coil in the buried electronic marker is arranged to be oriented vertically so that the location of the buried marker may be found directly beneath the position where the magnitude of the oscillatory magnetic field is detected to be at a maximum.
The magnetic flux density at a point on the axis of a circular coil consisting of N turns of radius a carrying current i at a distance x from the center of the coil may be determined using the Biot-Savart Law. It is given by:
      B    ⁡          (      x      )        =                              μ          0                ⁢        Ni            2        ⁢          (                        a          2                                      (                                          x                2                            +                              a                2                                      )                                3            2                              )          Where μ0 is the magnetic permeability of free space.    When x>>a, this simplifies to an inverse cube law:
      B    ⁡          (      x      )        =                    μ        0            ⁢              Nia        2                    2      ⁢                          ⁢              x        3            
A known system comprises a transmitting antenna and a receiving antenna. To determine the depth of cover to a buried electronic marker, the system is operated in a first position along the axis of the circular marker coil and the magnitude of the magnetic field radiated by the marker is measured and stored. Then the system is raised above the first position by a pre-defined distance and operated in a second position along the axis of the circular marker coil and the magnitude of the magnetic field radiated by the marker is measured and stored. Knowledge of the pre-defined distance enables the depth of the marker to be estimated using the stored values.
In this case the magnetic field received by the marker is reduced when the system is in the second position, as the transmitting antenna is further away from the marker by a distance s.
Let the radius of the transmitting antenna be denoted b and the distance along the axis of the transmit antenna coil be x. The magnitude of the magnetic field at the center of the marker coil due to the ampere turns Nitx in the transmit antenna coil is given by:
            B      m        ⁡          (      x      )        =                              μ          0                ⁢                  Ni          tx                            2        ⁢                                        ⁢          (                        b          2                                      (                                          x                2                            +                              b                2                                      )                                3            2                              )      When x>>b, this becomes:
            B      m        ⁡          (      x      )        =                    μ        0            ⁢              Ni        tx            ⁢              b        2                    2      ⁢                          ⁢              x        3            
Let the distance along the axis of the circular coil from a first position of the magnetic sensor to center of the circular coil=d. Let the distance along the axis of the circular coil from the first position of the magnetic sensor to the second position of the magnetic sensor=s. The magnitude of the magnetic field re-radiated by the marker is proportional to the magnitude of the magnetic field at the center of the marker coil due to the transmitter. Assuming the current in the transmitting antenna coil remains constant, the magnitudes of the magnetic fields detected by the sensor when the system is operated in the first and second positions respectively are:
            B      1        =                            Ka          2                          d          3                    ·                                    Ni            tx                    ⁢                      b            2                                    d          3                                B      2        =                            Ka          2                                      (                          d              +              s                        )                    3                    ·                                    Ni            tx                    ⁢                      b            2                                                (                          d              +              s                        )                    3                    Where K is a constant of proportionality representing the re-radiation efficiency of the marker. Find the ratio R of the magnitudes of the magnetic fields detected by the magnetic sensor in the first and second positions, then solve for the depth d:
      R    =                  B        1                    B        2                  R    =                                        Ka            2                                d            3                          ·                                            Ni              tx                        ⁢                          b              2                                            d            3                                                            Ka            2                                              (                              d                +                s                            )                        3                          ·                                            Ni              tx                        ⁢                          b              2                                                          (                              d                +                s                            )                        3                                    R    =                            (                      d            +            s                    )                6                    d        6                  R    =                  (                              d            +            s                    d                )            6      Solving for depth, d this gives:
  d  =      s          (                        R                      1            ⁢                          /                        ⁢            6                          -        1            )      
The sensitivity of the depth estimation to errors in the measurement of the ratio R may be determined by finding the derivative of depth d with respect to ratio R. For the system above, this is given by:
      S    1    =            -              1        6              ·          s              [                                            (                                                R                                      1                    ⁢                                          /                                        ⁢                    6                                                  -                1                            )                        2                    ·                      R                          5              ⁢                              /                            ⁢              6                                      ]            
As the depth equation contains an inverse sixth power, this system is highly sensitive to errors in determining R and suffers the disadvantage of requiring the user to lift the apparatus through a predefined distance s, introducing a further source of error.